Square Root of 448
2026-02-28 19:07 Diff
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Last updated on August 5, 2025

If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The square root is used in fields like vehicle design, finance, etc. Here, we will discuss the square root of 448.

What is the Square Root of 448?

The square root is the inverse of the square of the number. 448 is not a perfect square. The square root of 448 is expressed in both radical and exponential form. In the radical form, it is expressed as √448, whereas (448)^(1/2) in the exponential form. √448 ≈ 21.166, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 448

The prime factorization method is typically used for perfect square numbers. However, for non-perfect square numbers, methods like the long-division method and the approximation method are used. Let us now learn the following methods: 

  • Prime factorization method 
  • Long division method 
  • Approximation method

Square Root of 448 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 448 is broken down into its prime factors:

Step 1: Finding the prime factors of 448. Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 7: 2^5 x 7.

Step 2: Now we found the prime factors of 448. The second step is to make pairs of those prime factors. Since 448 is not a perfect square, the digits of the number can’t be grouped into complete pairs.

Therefore, calculating 448 using prime factorization is not straightforward.

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Square Root of 448 by Long Division Method

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step:

Step 1: To begin with, we need to group the numbers from right to left. In the case of 448, we need to group it as 48 and 4.

Step 2: Now we need to find n whose square is 4. We can say n is ‘2’ because 2 x 2 is equal to 4. Now the quotient is 2 after subtracting 4 from 4, the remainder is 0.

Step 3: Now let us bring down 48, which is the new dividend. Add the old divisor with the same number, 2 + 2, to get 4, which will be our new divisor.

Step 4: The new divisor will be part of finding 4n ≤ 48. Let us consider n as 1, now 4 x 1 = 4.

Step 5: Subtract 48 from 4; the difference is 44, and the quotient is 21.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 4400.

Step 7: Now we need to find a new divisor that is 42 because 421 ✖ 9 = 3789.

Step 8: Subtracting 3789 from 4400, we get the result 611.

Step 9: Now the quotient is 21.1.

Step 10: Continue these steps until we get a precise enough value.

So the square root of √448 ≈ 21.166.

Square Root of 448 by Approximation Method

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 448 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √448. The smallest perfect square less than 448 is 441, and the largest perfect square greater than 448 is 484. √448 falls somewhere between 21 and 22.

Step 2: Now we need to apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Going by the formula (448 - 441) ÷ (484 - 441) = 0.16.

Using the formula, we identified the decimal value.

The next step is adding the value we got initially to the decimal number, which is 21 + 0.16 = 21.16, so the square root of 448 is approximately 21.16.

Common Mistakes and How to Avoid Them in the Square Root of 448

Students often make mistakes while finding the square root, like forgetting about the negative square root or skipping steps in the long division method. Let's look at a few common mistakes in detail.

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Problem 1

Can you help Max find the area of a square box if its side length is given as √448?

Okay, lets begin

The area of the square is approximately 448 square units.

Explanation

The area of the square = side².

The side length is given as √448.

Area of the square = side² = √448 x √448 = 448.

Therefore, the area of the square box is 448 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 448 square feet is built; if each of the sides is √448, what will be the square feet of half of the building?

Okay, lets begin

224 square feet.

Explanation

We can just divide the given area by 2 as the building is square-shaped.

Dividing 448 by 2 gives us 224.

So half of the building measures 224 square feet.

Well explained 👍

Problem 3

Calculate √448 x 5.

Okay, lets begin

Approximately 105.83.

Explanation

The first step is to find the square root of 448, which is approximately 21.166, and then multiply 21.166 by 5.

So 21.166 x 5 ≈ 105.83.

Well explained 👍

Problem 4

What will be the square root of (441 + 7)?

Okay, lets begin

The square root is approximately 21.33.

Explanation

To find the square root, we need to find the sum of (441 + 7). 441 + 7 = 448, and then √448 ≈ 21.166.

Therefore, the square root of (441 + 7) is approximately 21.166.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √448 units and the width ‘w’ is 38 units.

Okay, lets begin

We find the perimeter of the rectangle as approximately 118.33 units.

Explanation

Perimeter of the rectangle = 2 × (length + width).

Perimeter = 2 × (√448 + 38) ≈ 2 × (21.166 + 38) = 2 × 59.166 ≈ 118.33 units.

Well explained 👍

FAQ on Square Root of 448

1.What is √448 in its simplest form?

The prime factorization of 448 is 2 x 2 x 2 x 2 x 2 x 7, so the simplest form of √448 = √(2^5 x 7).

2.Mention the factors of 448.

Factors of 448 are 1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 224, and 448.

3.Calculate the square of 448.

We get the square of 448 by multiplying the number by itself, that is 448 x 448 = 200704.

4.Is 448 a prime number?

5.448 is divisible by?

448 has many factors; those are 1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 224, and 448.

Important Glossaries for the Square Root of 448

  • Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, that is √16 = 4.
  • Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
  • Principal square root: A number has both positive and negative square roots; however, the positive square root is used more often due to its applications in the real world. This is known as the principal square root.
  • Prime factorization: The process of breaking down a number into its prime factors. For example, the prime factorization of 448 is 2^5 x 7.
  • Approximation method: A technique used to estimate the square root of a non-perfect square by identifying its closest perfect squares.

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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: He loves to play the quiz with kids through algebra to make kids love it.